6.6 Auxiliary Basis (Resolution of the Identity) MP2 Methods 6.6.6 Orbital-Optimized MP2 6.6.8 RI-TRIM MP2 Energies

6.6.7 Brueckner CC2

(September 1, 2024)

Brueckner orbitals (BOs) are shown to largely ameliorate the artificial symmetry breaking that occurs at the Hartree-Fock (HF) level. ⁷³⁹ Lee J., Head-Gordon M.
Phys. Chem. Chem. Phys.
(2019), 21, pp. 47638. Link This can lead to improved results even in higher order wavefunction theories that typically use HF orbitals as a starting point. Unfortunately, the cheapest traditional Brueckner theory, Brueckner doubles (BD), is still quite computationally demanding. Orbital optimized MP2 (OOMP2) was proposed as a low scaling approximation to BD ⁸⁰⁵ Lochan R. C., Head-Gordon M.
J. Chem. Phys.
(2007), 126, pp. 164101. Link due to the similarity of orbital optimized coupled cluster doubles (OOCCD) and BD. Another possible source of approximate BD orbitals is Bruckner CC2 (BCC2). ³⁰ Akinaga Y., Kawashima Y., Ten-no S.
Chem. Phys. Lett.
(2011), 506, pp. 276. Link

Rather than optimizing the energy as in OOMP2, BCC2 optimizes the orbitals in order to reduce the CC2 t ${}_{1}$ amplitudes to 0. In the absence of t ${}_{1}$ amplitudes, the CC2 doubles amplitudes are exactly the same as MP2, and the singles amplitude equations are as follows:

\Omega_{i}^{a}=f_{ai}+\sum_{jb}f_{jb}t_{ij}^{ab}+\frac{1}{2}\sum_{jbc}\langle ja% ||cb\rangle t_{ij}^{bc}+\frac{1}{2}\sum_{jkb}\langle jk||bi\rangle t_{jk}^{ab}

(6.24)

BCC2 has a computational cost scaling as $\mathcal{O}(o^{2}v^{3})$ - a significant improvement over OOCCD/BD, and can greatly improve the quality of orbitals over HF in open-shell molecules due to the reduction of artificial spin contamination.

Example 6.15 Example of BCC2 methodology

$molecule
   1 2
   F
   H 1 1.001
$end

$rem
   UNRESTRICTED      TRUE
   JOBTYPE           SP
   EXCHANGE          HF
   GEN_SCFMAN_FINAL  TRUE
   DO_BCC2           3                run BCC2
   SCF_ALGORITHM     DIIS
   SCF_GUESS         GWH
   BASIS             sto-3g
   AUX_BASIS         rimp2-vdz
   SCF_CONVERGENCE   8
   THRESH            14
   PURECART          1111
   INTEGRAL_SYMMETRY FALSE
$end

As CC2 is a perturbative approach similar to MP2, breakdowns occur in the presence of orbital degeneracy. While BCC2 does not explicitly optimize the energy and thereby drive orbitals toward degeneracy, results are still severely hindered when orbital energy differences become small. The $\kappa$ regularizer originally proposed by Joonho Lee and Martin Head-Gordon ⁷³⁸ Lee J., Head-Gordon M.
J. Chem. Theory Comput.
(2018), 14, pp. 5203. Link can be used in to further improve the results of BCC2 in this case. The regularizer is applied by modifying the t amplitudes as follows:

t_{ij}^{ab}=\frac{-\langle ab||ij\rangle}{\Delta_{ij}^{ab}}\left(1-\exp(-% \kappa\Delta_{ij}^{ab})\right)^{2}

(6.25)

As in $\kappa$ -OOMP2, the parameter $\kappa$ sets the regularization strength, with $\kappa=0$ yielding HF, and $\kappa=\infty$ yielding unregularized BCC2. A value of $\kappa=1.2$ $E_{h}^{-1}$ is suggested for most applications. $\kappa$ -BCC2 runs through libgscf and libgmbpt. Currently, DIIS should be used to converge BCC2 orbitals, as the singles residual is not an orbital gradient and cannot be used with gradient-based algorithms. The BCC2 code can currently handle restricted (R) and unrestricted (U) orbital types.

Summary of rem variables relevant to run $\kappa$ -BCC2:

CORRELATION	None (default)
JOBTYPE	sp (default) single point energy evaluation
BASIS	user’s choice (standard or user-defined: GENERAL or MIXED)
GEN_SCFMAN_FINAL	TRUE
SCF_ALGORITHM	DIIS (gradient based algorithms currently unsupported)
AUX_BASIS	corresponding auxiliary basis (standard or user-defined:
	AUX_GENERAL or AUX_MIXED)
DO_BCC2	3 (run BCC2)
REGULARIZED_BCC2	0 (no regularizer; default)
	1 ( $\delta$ -regularizer)
	2 ( $\kappa$ -regularizer; recommended)
	3 ( $\sigma$ -regularizer)
REG_PARAMETER	regularization parameter multiplied by 1e ${}^{3}$ ; no default
	1200 (Recommended value for $\kappa$ -BCC2)
N_FROZEN_CORE	0 (frozen core currently unsupported)
N_FROZEN_VIRTUAL	0 (frozen core currently unsupported)
DO_S2	0 (default)
	1 (Compute $\langle\hat{S}^{2}\rangle$ at the MP2 level)

Example 6.16 Example of $\kappa$ -BCC2 with the R orbital type applied to a water dimer

$molecule
   0  1
   O    -2.766559046    0.187082886    0.566917837
   H    -3.696304300    1.179189102   -0.642506882
   H    -3.395837846   -1.509891173    0.389283582
   O     2.587035064    0.275900014   -0.746441819
   H     3.579141280    0.918406897    0.633058252
   H     0.852266482    0.311804811   -0.156847268
$end

$rem
   EXCHANGE          hf
   BASIS             cc-pvdz
   AUX_BASIS_CORR    rimp2-cc-pvdz
   THRESH            14
   INPUT_BOHR        true
   SCF_CONVERGENCE   8
   MAXSCF            1000
   SCF_GUESS         sad
   GEN_SCFMAN_FINAL  true
   UNRESTRICTED      false    use restricted
   DO_BCC2           3        run BCC2
   REGULARIZED_O2    2        use kappa-regularizer
   REG_VARIABLE      1450     set kappa = 1.45
   INTEGRAL_SYMMETRY false
$end

Example 6.17 Example of $\kappa$ -BCC2 with the U orbital type applied to an OH radical

$molecule
0  2
O -2.766559046 0.187082886 0.566917837
H -3.6963043 1.179189102 -0.642506882
$end

$rem
   EXCHANGE          hf
   BASIS             cc-pvdz
   AUX_BASIS_CORR    rimp2-cc-pvdz
   THRESH            14
   INPUT_BOHR        true
   SCF_CONVERGENCE   8
   MAXSCF            1000
   SCF_GUESS         sad
   GEN_SCFMAN_FINAL  true
   UNRESTRICTED      true   ! use unrestricted
   DO_BCC2           3      ! run BCC2
   REGULARIZED_O2    2      ! use kappa-regularizer
   REG_VARIABLE      1450   ! set kappa = 1.45
   INTEGRAL_SYMMETRY false
$end

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6.6.7 Brueckner CC2